October 2017, 22(8): 3063-3077. doi: 10.3934/dcdsb.2017163

Integral conditions for nonuniform $μ$-dichotomy on the half-line

1. 

Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilh˜ã, Portugal

2. 

Department of Mathematics, "Politehnica" University of Timişoara, Victoriei Square 2,300006 Timişoara, Romania

3. 

Academy of Romanian Scientists, Independenţei 54,050094 Bucharest, Romania

Received  June 2016 Revised  January 2017 Published  June 2017

We give necessary integral conditions and sufficient ones for the existence of a general concept of $μ$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $μ$-dichotomies using these Lyapunov functions.

Citation: António J.G. Bento, Nicolae Lupa, Mihail Megan, César M. Silva. Integral conditions for nonuniform $μ$-dichotomy on the half-line. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3063-3077. doi: 10.3934/dcdsb.2017163
References:
[1]

M. G. Babuţia, M. Megan and I. -L. Popa, On $(h, k)$ -dichotomies for nonautonomous linear difference equations in Banach spaces, Int. J. Differ. Equ. , (2013), Art. ID 761680, 7 pages, URL http://dx.doi.org/10.1155/2013/761680

[2]

L. BarreiraJ. Chu and C. Valls, Lyapunov functions for general nonuniform dichotomies, Milan J. Math., 81 (2013), 153-169. doi: 10.1007/s00032-013-0198-y.

[3]

L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst. , 22 (2008), 509–528, URL http://dx.doi.org/10.3934/dcds.2008.22.509

[4]

L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, vol. 1926 of Lecture Notes in Mathematics, Springer, Berlin, 2008, URL http://dx.doi.org/10.1007/978-3-540-74775-8 doi: 10.1007/978-3-540-74775-8.

[5]

L. Barreira and C. Valls, Polynomial growth rates, Nonlinear Anal. , 71 (2009), 5208–5219, URL http://dx.doi.org/10.1016/j.na.2009.04.005

[6]

L. Barreira and C. Valls, Quadratic Lyapunov functions and nonuniform exponential dichotomies, J. Differential Equations, 246 (2009), 1235–1263, URL http://dx.doi.org/10.1016/j.jde.2008.06.008

[7]

A. J. G. Bento and C. Silva, Nonautonomous equations, generalized dichotomies and stable manifolds, ArXiv e-prints. URL http://arxiv.org/abs/0905.4935

[8]

A. J. G. Bento and C. Silva, Stable manifolds for nonuniform polynomial dichotomies, J. Funct. Anal. , 257 (2009), 122–148, URL http://dx.doi.org/10.1016/j.jfa.2009.01.032

[9]

A. J. G. Bento and C. M. Silva, Stable manifolds for non-autonomous equations with nonuniform polynomial dichotomies, Q. J. Math. , 63 (2012), 275–308, URL http://dx.doi.org/10.1093/qmath/haq047

[10]

A. J. G. Bento and C. M. Silva, Generalized nonuniform dichotomies and local stable manifolds, J. Dynam. Differential Equations, 25 (2013), 1139–1158, URL http://dx.doi.org/10.1007/s10884-013-9331-4

[11]

T. Burton and L. Hatvani, Stability theorems for nonautonomous functional-differential equations by Liapunov functionals, Tohoku Math. J. , (2) 41 (1989), 65–104, URL http://dx.doi.org/10.2748/tmj/1178227868

[12]

T. A. Burton and L. Hatvani, On nonuniform asymptotic stability for nonautonomous functional-differential equations, Differential Integral Equations, 3 (1990), 285–293, URL http://projecteuclid.org/euclid.die/1371586144

[13]

X. Chang, J. Zhang and J. Qin, Robustness of nonuniform $(μ, ν)$ -dichotomies in Banach spaces, J. Math. Anal. Appl. , 387 (2012), 582–594, URL http://dx.doi.org/10.1016/j.jmaa.2011.09.026

[14]

C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, vol. 70 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999. URL http://dx.doi.org/10.1090/surv/070 doi: 10.1090/surv/070.

[15]

R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. , 3 (1972), 428–445, URL http://dx.doi.org/10.1137/0503042

[16]

L. Hatvani, On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals, Trans. Amer. Math. Soc. , 354 (2002), 3555–3571, URL http://dx.doi.org/10.1090/S0002-9947-02-03029-5

[17]

T. Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math. , 63 (1934), 99–141, URL http://dx.doi.org/10.1007/BF02547352

[18]

N. Lupa and M. Megan, Exponential dichotomies of evolution operators in Banach spaces, Monatsh. Math. , 174 (2014), 265–284, URL http://dx.doi.org/10.1007/s00605-013-0517-y

[19]

A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521–790, translated by A. T. Fuller from Édouard Davaux's French translation (1907) ´ of the 1892 Russian original, With an editorial (historical introduction) by Fuller, a biography of Lyapunov by V. I. Smirnov, and the bibliography of Lyapunov's works collected by J. F. Barrett, Lyapunov centenary issue. URL http://dx.doi.org/10.1080/00207179208934253

[20]

A. D. Maǐzel0, On stability of solutions of systems of differential equations, Ural. Politehn. Inst. Trudy, 51 (1954), 20-50.

[21]

M. Megan, On $(h,k)$ -dichotomy of evolution operators in Banach spaces, Dynam. Systems Appl., 5 (1996), 189-195.

[22]

M. Megan and C. Buşe, Dichotomies and Lyapunov functions in Banach spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 37 (1993), 103-114.

[23]

Y. A. Mitropolsky, A. M. Samoilenko and V. L. Kulik, Dichotomies and Stability in Nonautonomous Linear Systems, vol. 14 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, 2003.

[24]

R. Naulin and M. Pinto, Roughness of $(h, k)$ -dichotomies, J. Differential Equations, 118 (1995), 20–35, URL http://dx.doi.org/10.1006/jdeq.1995.1065

[25]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. , 32 (1930), 703–728, URL http://dx.doi.org/10.1007/BF01194662

[26]

Y. Pesin, Families of invariant manifolds that corresponding to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. , 40 (1976), 1332–1379, (Russian) English transl. Math. USSR-Izv. , 10 (1976), 1261–1305, URL http://dx.doi.org/10.1070/IM1976v010n06ABEH001835

[27]

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55–112, (Russian) English transl. Russ. Math. Surv. , 32 (1977), 55–114, URL http://dx.doi.org/10.1070/RM1977v032n04ABEH001639

[28]

Y. Pesin, Geodesic flows in closed Riemannian manifolds without focal points, Izv. Akad. Nauk SSSR Ser. Mat. , 41 (1977), 1252–1288, (Russian) English transl. Math. USSR-Izv. , 41 (1977), 1195–1228, URL http://dx.doi.org/doi:10.1070/IM1977v011n06ABEH001766

[29]

M. Pinto, Discrete dichotomies, Comput. Math. Appl. , 28 (1994), 259–270, URL http://dx.doi.org/10.1016/0898-1221(94)00114-6

[30]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems Vol. 2002 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, URL http://dx.doi.org/10.1007/978-3-642-14258-1 doi: 10.1007/978-3-642-14258-1.

[31]

C. Preda, P. Preda and A. Craciunescu, A version of a theorem of R. Datko for nonuniform exponential contractions, J. Math. Anal. Appl. , 385 (2012), 572–581, URL http://dx.doi.org/10.1016/j.jmaa.2011.06.082

[32]

P. Preda and M. Megan, Exponential dichotomy of evolutionary processes in Banach spaces, Czechoslovak Math. J. , 35 (1985), 312–323, URL http://dml.cz/handle/10338.dmlcz/102019

[33]

A. L. Sasu, M. G. Babut¸ia and B. Sasu, Admissibility and nonuniform exponential dichotomy on the half-line, Bull. Sci. Math. , 137 (2013), 466–484, URL http://dx.doi.org/10.1016/j.bulsci.2012.11.002

[34]

B. Sasu, Integral conditions for exponential dichotomy: A nonlinear approach, Bull. Sci. Math. , 134 (2010), 235–246, URL http://dx.doi.org/10.1016/j.bulsci.2009.06.006

show all references

References:
[1]

M. G. Babuţia, M. Megan and I. -L. Popa, On $(h, k)$ -dichotomies for nonautonomous linear difference equations in Banach spaces, Int. J. Differ. Equ. , (2013), Art. ID 761680, 7 pages, URL http://dx.doi.org/10.1155/2013/761680

[2]

L. BarreiraJ. Chu and C. Valls, Lyapunov functions for general nonuniform dichotomies, Milan J. Math., 81 (2013), 153-169. doi: 10.1007/s00032-013-0198-y.

[3]

L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst. , 22 (2008), 509–528, URL http://dx.doi.org/10.3934/dcds.2008.22.509

[4]

L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, vol. 1926 of Lecture Notes in Mathematics, Springer, Berlin, 2008, URL http://dx.doi.org/10.1007/978-3-540-74775-8 doi: 10.1007/978-3-540-74775-8.

[5]

L. Barreira and C. Valls, Polynomial growth rates, Nonlinear Anal. , 71 (2009), 5208–5219, URL http://dx.doi.org/10.1016/j.na.2009.04.005

[6]

L. Barreira and C. Valls, Quadratic Lyapunov functions and nonuniform exponential dichotomies, J. Differential Equations, 246 (2009), 1235–1263, URL http://dx.doi.org/10.1016/j.jde.2008.06.008

[7]

A. J. G. Bento and C. Silva, Nonautonomous equations, generalized dichotomies and stable manifolds, ArXiv e-prints. URL http://arxiv.org/abs/0905.4935

[8]

A. J. G. Bento and C. Silva, Stable manifolds for nonuniform polynomial dichotomies, J. Funct. Anal. , 257 (2009), 122–148, URL http://dx.doi.org/10.1016/j.jfa.2009.01.032

[9]

A. J. G. Bento and C. M. Silva, Stable manifolds for non-autonomous equations with nonuniform polynomial dichotomies, Q. J. Math. , 63 (2012), 275–308, URL http://dx.doi.org/10.1093/qmath/haq047

[10]

A. J. G. Bento and C. M. Silva, Generalized nonuniform dichotomies and local stable manifolds, J. Dynam. Differential Equations, 25 (2013), 1139–1158, URL http://dx.doi.org/10.1007/s10884-013-9331-4

[11]

T. Burton and L. Hatvani, Stability theorems for nonautonomous functional-differential equations by Liapunov functionals, Tohoku Math. J. , (2) 41 (1989), 65–104, URL http://dx.doi.org/10.2748/tmj/1178227868

[12]

T. A. Burton and L. Hatvani, On nonuniform asymptotic stability for nonautonomous functional-differential equations, Differential Integral Equations, 3 (1990), 285–293, URL http://projecteuclid.org/euclid.die/1371586144

[13]

X. Chang, J. Zhang and J. Qin, Robustness of nonuniform $(μ, ν)$ -dichotomies in Banach spaces, J. Math. Anal. Appl. , 387 (2012), 582–594, URL http://dx.doi.org/10.1016/j.jmaa.2011.09.026

[14]

C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, vol. 70 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999. URL http://dx.doi.org/10.1090/surv/070 doi: 10.1090/surv/070.

[15]

R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. , 3 (1972), 428–445, URL http://dx.doi.org/10.1137/0503042

[16]

L. Hatvani, On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals, Trans. Amer. Math. Soc. , 354 (2002), 3555–3571, URL http://dx.doi.org/10.1090/S0002-9947-02-03029-5

[17]

T. Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math. , 63 (1934), 99–141, URL http://dx.doi.org/10.1007/BF02547352

[18]

N. Lupa and M. Megan, Exponential dichotomies of evolution operators in Banach spaces, Monatsh. Math. , 174 (2014), 265–284, URL http://dx.doi.org/10.1007/s00605-013-0517-y

[19]

A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521–790, translated by A. T. Fuller from Édouard Davaux's French translation (1907) ´ of the 1892 Russian original, With an editorial (historical introduction) by Fuller, a biography of Lyapunov by V. I. Smirnov, and the bibliography of Lyapunov's works collected by J. F. Barrett, Lyapunov centenary issue. URL http://dx.doi.org/10.1080/00207179208934253

[20]

A. D. Maǐzel0, On stability of solutions of systems of differential equations, Ural. Politehn. Inst. Trudy, 51 (1954), 20-50.

[21]

M. Megan, On $(h,k)$ -dichotomy of evolution operators in Banach spaces, Dynam. Systems Appl., 5 (1996), 189-195.

[22]

M. Megan and C. Buşe, Dichotomies and Lyapunov functions in Banach spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 37 (1993), 103-114.

[23]

Y. A. Mitropolsky, A. M. Samoilenko and V. L. Kulik, Dichotomies and Stability in Nonautonomous Linear Systems, vol. 14 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, 2003.

[24]

R. Naulin and M. Pinto, Roughness of $(h, k)$ -dichotomies, J. Differential Equations, 118 (1995), 20–35, URL http://dx.doi.org/10.1006/jdeq.1995.1065

[25]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. , 32 (1930), 703–728, URL http://dx.doi.org/10.1007/BF01194662

[26]

Y. Pesin, Families of invariant manifolds that corresponding to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. , 40 (1976), 1332–1379, (Russian) English transl. Math. USSR-Izv. , 10 (1976), 1261–1305, URL http://dx.doi.org/10.1070/IM1976v010n06ABEH001835

[27]

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55–112, (Russian) English transl. Russ. Math. Surv. , 32 (1977), 55–114, URL http://dx.doi.org/10.1070/RM1977v032n04ABEH001639

[28]

Y. Pesin, Geodesic flows in closed Riemannian manifolds without focal points, Izv. Akad. Nauk SSSR Ser. Mat. , 41 (1977), 1252–1288, (Russian) English transl. Math. USSR-Izv. , 41 (1977), 1195–1228, URL http://dx.doi.org/doi:10.1070/IM1977v011n06ABEH001766

[29]

M. Pinto, Discrete dichotomies, Comput. Math. Appl. , 28 (1994), 259–270, URL http://dx.doi.org/10.1016/0898-1221(94)00114-6

[30]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems Vol. 2002 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, URL http://dx.doi.org/10.1007/978-3-642-14258-1 doi: 10.1007/978-3-642-14258-1.

[31]

C. Preda, P. Preda and A. Craciunescu, A version of a theorem of R. Datko for nonuniform exponential contractions, J. Math. Anal. Appl. , 385 (2012), 572–581, URL http://dx.doi.org/10.1016/j.jmaa.2011.06.082

[32]

P. Preda and M. Megan, Exponential dichotomy of evolutionary processes in Banach spaces, Czechoslovak Math. J. , 35 (1985), 312–323, URL http://dml.cz/handle/10338.dmlcz/102019

[33]

A. L. Sasu, M. G. Babut¸ia and B. Sasu, Admissibility and nonuniform exponential dichotomy on the half-line, Bull. Sci. Math. , 137 (2013), 466–484, URL http://dx.doi.org/10.1016/j.bulsci.2012.11.002

[34]

B. Sasu, Integral conditions for exponential dichotomy: A nonlinear approach, Bull. Sci. Math. , 134 (2010), 235–246, URL http://dx.doi.org/10.1016/j.bulsci.2009.06.006

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